Physics 311General Relativity Lecture 15: Metrics and curved space
Today’s lecture plan • Flat spacetime of Special Relativity. • Solving Einstein Field Equation for empty space – the “vacuum solution” • Schwarzschild metric
A look back • In Special Relativity the spacetime is said to be “flat”, it has no “curvature”. What do we mean when we say “the spacetime is flat”, “the spacetime has no curvature”? • We mean that the path of a free particle is a straight line, and that the square of the interval is a linear combination of the space and time components squared: • ds2 = dt2 – (dx2 + dy2 + dz2) • (in the system of units where c = 1) • This is a lot like the Euclidean geometry, which is also flat. We’ve alluded to a non-Euclidean geometry in the last lecture; we’ll soon see how it comes to be.
The Minkowski metric • We can write the expression for the interval in the matrix form: • 1 0 0 0 dt • 0 i 0 0 dx • 0 0 i 0 dy • 0 0 0 i dz • ( so that ds2 = dt2 - dx2 - dy2 - dz2) • This matrix – a very simple matrix indeed – defines the metric of Special Relativity, the Minkowski metric. It is simple yet powerful; it completely describes the spacetime of Special Relativity. ds = x
Einstein Field Equation: another dissection • Generally speaking, Einstein field equation • Gmn = 8pTmn • is a set of 10 (16 components in each tensor, minus 6 due to symmetry, and if you’re smart enough, minus 4 more) coupled elliptic-hyperbolic nonlinear partial differential equations for the metric components. • Just so that we are clear on definitions: • “coupled” – each differential equation contains multiple terms; equations cannot be solved individually. • “elliptic-hyperbolic” – determinants of sub-matrices of the system of equations matrix are either positive or negative; never zero. • “nonlinear” – dependent on nonlinear function of metric components • “partial differential equation” – an equation containing partial derivatives of functions, for example 2f(x,y,z)/xy • “metric components” – components of the metric tensor gmn
Rmn R Einstein Field Equation expanded • From “simple” form: Gmn = 8pTmn ...to rather more complex, expanded form... [h(Gmhn ) - n(Ghhm ) + Ghhl Gmln - Gmlh Gnhl] - - ½ gab[h(Gahb ) - b(Ghha ) + Ghhl Galb - Galh Gbhl] gmn= 8pTmn Here, Gagb= ½ gdg[ b(gad) + a(gbd) - d(gab)] are Christoffel symbols of 2nd kind – tensor-like objects derived from Riemann metric g; a = (/xa) denotes partial derivative with respect to variable xa; and gab is the metric tensor – roughly speaking, the function that tells us how to compute distances between points in a given space: ds2 = Sgabdxadxb
Back to Minkowski metric • The matrix in the expression for the interval is nothing more, nor less, than the Minkowski metric tensor gab • 1 0 0 0 dt • 0 i 0 0 dx • 0 0 i 0 dy • 0 0 0 i dz • So we actually know one metric tensor already – it’s not too scary at all! ds = x
If you sit down and write down the Ricci tensor for a general case of a 2-dimensional space with axial symmetry, you would get something like this:
... and just a little bit more. This is a general expression for Ricci tensor Rmn in only two dimensions, with axial symmetry. (From Larry Smarr, Univ. of Illinois) Just try to imagine all of three dimensions of space plus one of time!
Special case: vacuum • I haven’t said anything about the energy-stress tensor Tmn yet. Well, here’s an example of this tensor: • Tmn= 0 • This special case is called “vacuum”, and corresponding solutions for the metric gmn are called “vacuum solutions”. In this case, we have R = 0. • Wait – what solutions? If we set Tmn to zero, wouldn’t our metric be just zero as well? • Not really! The field equation now has form: • Rmn = 0 • But the left-hand side is a complicated mess of derivatives of the metric. There can be many solutions for this “vacuum” equation, including several exact analytic solution. These different solutions arise from different symmetries we impose on the metric.
Minkowski metric (one last time) • Minkowski metric is one of the vacuum solutions for a space that has perfect symmetry – a space that is: • - uniform, so thatg(t,x,y,z) = g(t,x+Dx,y,z) (also true for y and z) • - isotropic, so thatg(t,x,y,z) = g(t,-x,y,z) (also true for y and z) • and a time that is: • - uniform, so thatg(t,x,y,z) = g(t+Dt,x,y,z)
Schwarzschild Vacuum Solution • Another important metric, first to be explicitly solved only weeks after Einstein published his General Relativity paper is 1915, is called Schwarzschild metric, named after the man who solved it. • This solution assumes spherical symmetry of space, as around an isolated star. • How is this “vacuum” if there is a star?! There’s mass, thus there is energy, and there must be stress somehow, so tensor Tmn must be nonzero! • The keyword is “around” – the solution is for the metric of empty space (also known as “vacuum”) surrounding a spherically-symmetric massive object. Karl Schwarzschild
Derivation of Schwarzschild solution. 1. Assumptions and notation • We start by defining our assumptions and notation. • 1. The coordinates are (t, r, q, f) – time + spherical coordinate system. We call these coordinates xm, with m = 1...4. • 2. Spherical symmetry: metric components are unchanged under r → -r,q → -q and f → -f. • 3. Spacetime is static, i.e. all metric components are independent of time: (gmn/t) = 0; this also means that spacetime is invariant under time reversal. • 4. We are looking for vacuum solution Tmn = 0, with R = 0. • What we need to solve then is: • Rmn = 0
Derivation of Schwarzschild solution. 2. Diagonalizing • The requirements that metric be time-independent, and symmetric with respect to rotations, allow us to diagonalize the matrix: • 1. Time-reversal symmetry: (t, r, q, f) →(-t, r, q, f) must conserve components of g. The components of the 1st column of the metric, gm1 (m ≠ 1), transforms under time reversal as: gm1 → -gm1 • Since we demand that gm1 = gm1, then gm1 = 0 for (m ≠ 1). • 2. Same reasoning for r, q and f – symmetries leads to all other non-diagonal (i.e. m ≠ n) metric components to vanish. • Thus, the sought metric has the form: • ds2 = g11dt2 + g22dr2 + g33dq2 + g44df2
Derivation of Schwarzschild solution. 3. Simplifying • On a sphere of constant radius, and at constant time, the only spherically-symmetric combination of dq2 and df2 is C(r)(dq2 + sin2qdf2), where C(r) is (a yet unknown) function of radius coordinate only. This expression above is simply the element of a spherical surface. • For constant t, q and f (i.e. on the radial line) metric should only depend on the radius coordinate r – again, to conserve the spherical symmetry. That means that the metric components for time and radius, g11 and g22, must be functions of r only. • This simplifies the metric even further, to: • ds2 = A(r)dt2 + B(r)dr2 + C(r)(dq2 + sin2qdf2)
Derivation of Schwarzschild solution. 4. Solving for components • First, we find the function C(r) by noticing that at a surface of constant radius r0 and at constant time, the separation can be written as: • ds2 = r02(dq2 + sin2qdf2) • Since this must hold true for all radial surfaces, i.e. for any r, the unknown function C(r) is simply r2, and the q and f – components of the metric are: • g33 = r2 • g44 = r2sin2q ds r0 r0
Derivation of Schwarzschild solution. 4.1. Solving for components • Functions A(r) and B(r) can be found by solving the Einstein field equation (what a surprise!). Only 4 equations remain non-trivial: • 4rAB – 2rr2BAB + rrArBB + rrB2A = 0 • rrAB + 2A2B – 2AB – rrBA = 0 • -2rr2BAB + rrArBB + rrB2A – 4rBA = 0 • (-2rr2BAB + rrArBB + rrB2A – 4rBAB)sin2q = 0 • Subtracting equations 1 and 3 leads to: • rAB + rBA = 0 A(r)B(r) = K (a non-zero, real constant) • Substituting into equation 2 we get: • rrA – A(1 – A) = 0 A(r) = K[1 + 1/(Sr)] = g11 • B(r) = [1 + 1/(Sr)]-1 = g22
Derivation of Schwarzschild solution. 5. Arriving at solution • Finally, we find the coefficients K and S in the weak-field approximation – i.e. far away from the gravitational source. At r → the spacetime must approach Minkowski spacetime, thus: g11 = K[1 + 1/(Sr)] → K K = c2 = 1 • Gravity must converge to Newtonian in the weak field. This lets us find the numerical value of the constant S: S = -c2/(2Gm) = -1/(2m) where m is the mass of the central body, and G is the gravitational constant. • The full Schwarzschild metric is: • ds2 = [1-(2m/r)]dt2 – (1-(2m/r))-1dr2 - r2dq2 - r2sin2qdf2
Schwarzschild spacetime • Schwarzschild spacetime has curvature that decreases with distance from the center. At infinity, Schwarzschild spacetime is identical to the flat Minkowski spacetime. • In the center of Schwarzschild metric, singularity is possible, leading to formation of a Schwarzschild (non-rotating) black hole.
Recap: • Einstein field equations can be explicitly solved for certain types of stress-energy tensor. These solutions are called spacetime metrics. • Special case of stress-energy tensor – the vacuum – leads to Minkowski and Schwarzschild spacetime (among many others). • Schwarzschild metric is fairly simple. We will mostly see its 3-dimentional (one time plus two space) case: • ds2 = [1 – (2m/r)]dt2 – [1 – (2m/r)]-1dr2 – r2df2